Problem: Find $\dfrac{d}{dx}(8\cdot3^x)$. Choose 1 answer: Choose 1 answer: (Choice A) A $8\cdot 3^{x-1}$ (Choice B) B $8\cdot 3^x\log_3(x)$ (Choice C) C $8\cdot 3^x\ln(x)$ (Choice D) D $8\cdot 3^x\ln(3)$
Solution: The expression to differentiate includes an exponential term. Remember that the derivative of the general exponential term $a^x$ (where $a$ is any positive constant) is $\ln(a)\cdot a^x$. Put another way, $\dfrac{d}{dx}(a^x)=\ln(a)\cdot a^x$. $\begin{aligned} &\phantom{=}\dfrac{d}{dx}(8\cdot3^x) \\\\ &=8\dfrac{d}{dx}(3^x) \\\\ &=8\cdot\ln(3)\cdot3^x \\\\ &=8\cdot3^x\ln(3) \end{aligned}$ In conclusion, $\dfrac{d}{dx}(8\cdot3^x)=8\cdot3^x\ln(3)$.